This
I have a 17 pound bow and i sometimes practise by shooting at someone else with a similar bow with the same arrow. I.e i shoot at him - he grabs it midair and shoots it back.
But if you have a bow used for any kind warfare (i used a 60 pound bow at archery for reference) it's damn near IMPOSSIBLE to move in time.
Well firstly, warbows were about 100lb draw weight, but draw weight has nothing to do with the arrow speed, it only determines the maximum and optimal arrow weight along with the 'throw' and springiness of the bow. High draw weight longbows were not too springy so they used heavy arrows instead to maximize the bow's killing power. Nontheless all medieval bows were pretty slow in terms of arrow speed and if someone would fire from 20m it's still very possible to dodge the arrow, but definently not catch it.
But back to topic, about chambering thrown projectiles. I don't think that in reality it would be far fetched idea to deflect or atleast reduce the damage done by say and throwing axe, they are really slow and when spinning in the air a really big target for any stick like thing in 90 degrees angle to it's trajectory. Sure, it still might hit you but it wouldn't prolly crush your skull.